ABSTRACT The statistical analysis of failure time data has been the cornerstone and plays an essential role in the design and analysis of various medical studies such as randomized clinical trials. A key feature of failure time data that separates them from all other types of data is censoring, which makes the analysis of failure time data unique and difficult and can occur in different forms. This proposal will investigate two fundamental problems in medical studies, treatment comparison and estimation of relative risks, with respect to several types of failure time data that involve interval-censoring or interval-censored failure time data. The analysis of such data has been attracting more and more attention among medical investigators and statistician including these in government agencies and pharmaceutical companies. Specifically, the proposed research consists of three aims and they are the development of appropriate and/or efficient statistical procedures for 1) treatment comparison and sample size calculation, 2) estimation of relative risks and regression analysis I, and 3) estimation of relative risks and regression analysis II. It is well-known that treatment comparison is perhaps the most basic and commonly required task in medical studies and for the design of such studies, a key element is the determination of the required sample size. In the case of interval-censored data, some comparison procedures have been proposed. However, most of them are limited in applications and more importantly, none of them can be used for sample size calculation. Aim 1 will develop new comparison procedures that allow and thus give formulas for the sample size calculation. Estimation of relative risks or more generally covariate effects is another common task in medical studies such as progression-free survival oncology studies. The proposed research will develop appropriate or efficient approaches to it when one faces various types of interval-censored data including clustered data. Aim 2 will focus on situations where the censoring can be regarded to be independent of the failure time variable under study, while aim 3 will deal with situations where the independence is not true. The statistical procedures or tools that will be developed in this proposed research will make the design of the concerned studies possible and help one to conduct correct and/or efficient analysis of them.